# zbMATH — the first resource for mathematics

On some discontinuous fixed point mappings in convex metric spaces. (English) Zbl 0814.47065
This article deals with nonlinear operators $$T$$ in metric linear spaces satisfying either the condition $d(Tx, Ty)\leq ad(x, y)+ (1- a)\max \{d(x, Tx), d(y, Ty), b[d(x, Ty)+ d(y, Tx)]\}$ with $$0\leq a< 1$$ and $$b\leq {1\over 2}- (1- a^ 2)/(10+ 6a^ 2)$$ or the condition $d(Tx, Ty)\leq ad(x, y)+ b[d(x, Ty)+ d(y, Tx)]+ c\max \{d(x, Tx), d(y, Ty)\}$ with $$0\leq a< 1$$, $$b$$, $$c\geq 0$$, $$a+ b> 0$$ and $$a+ b(5+ a^ 2)/(2+ a^ 2)+ c\leq 1$$. The main results are the fixed point theorems for such operators.
Reviewer: P.Zabreiko (Minsk)

##### MSC:
 47H10 Fixed-point theorems
Full Text:
##### References:
 [1] Lj. B. Ćirič: Generalized contractions and fixed-point theorems. Publ. Inst. Math. (Beograd) 26 (1971), 19-26. · Zbl 0234.54029 [2] Lj. B. Ćirič: On a common fixed point theorem of a Greguš type. Publ. Inst. Math. (Beograd) 63 (1991), no. 49, 174-178. · Zbl 0753.54023 [3] D. Delbosco, O. Ferrero and F. Rossati: Teoremi di punto fisso per applicazioni negli spazi di Banach. Boll. Un. Math. Ital. 2-A (1983), no. 6, 297-303. · Zbl 0532.47046 [4] M. L. Diviccaro, B. Fisher and S. Sessa: A common fixed point theorem of Greguš type. Publ. Math. Debrecen 34 (1987), no. 1-2, 83-89. · Zbl 0634.47051 [5] B. Fisher: Common fixed points on a Banach space. Chung Yuan J. 11 (1982), 19-26. [6] B. Fisher and S. Sessa: On a fixed point theorem of Greguš. Internat. J. Math. Math. Sci. 9 (1986), no. 1, 23-28. · Zbl 0597.47036 · doi:10.1155/S0161171286000030 · eudml:46091 [7] M. Greguš: A fixed point theorem in Banach space. Boll. Un. Mat. Ital. 5 (1980), no. 17-A, 193-198. · Zbl 0538.47035 [8] B. Y. Li: Fixed point theorems of nonexpansive mappings in convex metric spaces. Appl. Math. Mech. (English Ed) 10 (1989), no. 2, 183-188. · Zbl 0752.47022 · doi:10.1007/BF02014826 [9] R. N. Mukherjee and V. Verma: A note on a fixed point theorem of Greguš. Math. Japon 33 (1988), 745-749. · Zbl 0655.47047 [10] W. Takahashi: A convexity in metric space and nonexpansive mappings I. Kodai Math. Sem. Rep. 22 (1970), 142-149. · Zbl 0268.54048 · doi:10.2996/kmj/1138846111
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.